A small fly of mass 0.25 g is caught in a spider’s web. The web vibrates predominately with a frequency of 5 Hz. Find the value of the spring constant.
To find the value of the spring constant, we need to use the equation of motion for simple harmonic motion.
The equation for the period (T) of an object undergoing simple harmonic motion is given by:
T = 1 / f
Where:
- T is the period of the motion in seconds,
- f is the frequency of the motion in hertz.
In this case, we are given the frequency of the vibration, which is 5 Hz. Dividing this value by 1 will give us the period of the motion:
T = 1 / 5 Hz = 0.2 seconds.
The period of the motion corresponds to the time it takes for one complete cycle. In other words, the fly oscillates back and forth in the web with a period of 0.2 seconds.
Now, we can use another equation for simple harmonic motion:
T = 2π√(m / k)
Where:
- T is the period of the motion,
- π is a mathematical constant (approximately equal to 3.14),
- m is the mass of the object in kilograms,
- k is the spring constant in N/m.
We are given the mass of the fly as 0.25 g. To convert this to kilograms, we divide it by 1000:
m = 0.25 g / 1000 = 0.00025 kg.
Now we can rearrange the equation to solve for the spring constant:
k = (4π²m) / T²
Substituting the known values:
k = (4π² * 0.00025 kg) / (0.2 seconds)²
Calculating this expression:
k ≈ 98.87 N/m.
Therefore, the value of the spring constant is approximately 98.87 N/m.